PFV compressibility, not truly compressible?

Hi Robert,
Imagine a compressible fluid flowing in a pipe.
The fluid density is not the same at the inlet and outlet due to pressure gradient. Nevertheless you can always define locally, in one point along the pipe, a relation between the local pressure gradient and the local mass flux. In the stokes regime this relation is linear. At this point I did not need to assume incompessibility, parallel plates, circular cross section or anything else. Just local linearity vs. pressure gradient.

What you say (if I understand) leads me to consider that the coefficient of the linearity itself (conductivity) can be a function of the absolute pressure. I agree that the change of density may lead to a change of viscosity, which means a change of conductivity. This is the only way I see for the compressibility to have an influence on the flow property.

I would thus describe the current model as "approximating the viscosity by a constant value independent of absolute pressure".
This would certainly not work for gas storage, but I guess it is acceptable for most geomechanical fluids. Else it would need a more sophisticated solver since the problem with non-constant viscosity is non-linear.

Besides, I think it is truly compressible. :)

Bruno

Dear Bruno,

Thank you for the detailed response. At the pore scale of geomaterials, I believe the assumption of local linearity vs pressure gradient in the pore throat is a safe one since the pore throat is so short compared to the pores themselves.

Let's say we really want to incorporate compressibility into the pore throat without going full 3D Navier-Stokes. I believe one option is to discretize the pore throat and linearize the pressure gradient at each point along the discretization. The irony is palpable here, as per your post, this is essentially what we are already doing at a larger scale of fluid movement through the porous medium. But in my experience, the only way to approach a non-linear problem numerically is to linearize it in one way or another.

Cheers,

Robert

By sub-discretizing the flow path we could account for the gradient of
viscosity at the throat scale, indeed.
The fact is that even at the large scale we do not account for it. We have
a unique value of viscosity which is supposed to apply everywhere in the
problem. I think this is the only approximation.
Do you see another problem in relation with compressibility?
Bruno

On 6 June 2017 at 23:23, Robert Caulk <email address hidden>
wrote:

> Question #640093 on Yade changed:
> https://answers.launchpad.net/yade/+question/640093
>
> Status: Answered => Solved
>
> Robert Caulk confirmed that the question is solved:
> Dear Bruno,
>
> Thank you for the detailed response. At the pore scale of geomaterials,
> I believe the assumption of local linearity vs pressure gradient in the
> pore throat is a safe one since the pore throat is so short compared to
> the pores themselves.
>
> Let's say we really want to incorporate compressibility into the pore
> throat without going full 3D Navier-Stokes. I believe one option is to
> discretize the pore throat and linearize the pressure gradient at each
> point along the discretization. The irony is palpable here, as per your
> post, this is essentially what we are already doing at a larger scale of
> fluid movement through the porous medium. But in my experience, the only
> way to approach a non-linear problem numerically is to linearize it in
> one way or another.
>
>
> Cheers,
>
> Robert
>
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Yes, the constant viscosity is an ok assumption since we are assuming constant temperatures. If I decide to add thermal processes to yade, I would adjust viscosity as a function of temperature.

I need to run some small tests to decide if the compressibility along the pore throat will have much of an impact on the poroelastic behavior. If I do need to discretize the pore throat, I will also need to augment the mechanical coupling of the pore throat pressures to the DE...

This is all starting to sound time and consuming and computationally expensive :-)

> viscosity is an ok assumption since we are assuming constant temperatures

My point was that the change of viscosity by compressibility (not temperature) is what we neglect. Other than that I don't see why the computational scheme would be "not truly compressible". Do I miss something?
Which equations do you have in mind for compressible flow between parallel plates?
B

Ah, I think I understand now. You are saying that the only approximation is the constant viscosity. I was just saying the change of viscosity of geofluids as a function of pressure is small enough in the liquid phase at a constant temperature that it is OK to neglect it. This is why I say we are assuming constant temperature with our constant viscosity assumption. But this is likely what you were getting at when you mentioned that this model is inappropriate for gases earlier.

I also now realize that the change of fluid density at low mach numbers is negligible too, so it seems the scheme is actually accurately considering all possible compressibility given the aforementioned assumptions. I guess it would be futile to try and gain accuracy by considering these small changes to viscosity within the pore throats without considering full convective and conductive heat transfer in the system.

>Which equations do you have in mind for compressible flow between parallel plates?

Honestly, I need to spend time looking into the options. But my preliminary search was pretty bleak a couple weeks ago.

Thank you for taking the time to explain these details to me.

Best,

Robert